← Bring every length to one unit before multiplying, and convert back last · Surface Area and Volume of Solids

Bring every length to one unit before multiplying, and convert back last · 12 practice problems

6.G.A.25.MD.C.56.RP.A.3

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 0.5

The rectangular prism at the right has a volume of $3600000\ \text{cm}^3$ . Find the number that belongs in $\bigcirc$ .

Show solution

1 · Understandwhat's really being asked

A prism has edges 3 m, 240 cm and an unknown number of metres, and its volume is 3600000 cm3. We want the unknown, in metres.

Givens

One edge is 3 m and another is 240 cm.
The volume is 3600000 cm3.
The third edge is given in metres.

Unknowns

The number of metres in the third edge.

Constraints

The volume is in cubic centimetres, so the working has to be in centimetres.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #11 Work Backwards#13 Convert to Algebra

Three lengths in two units and a volume in a third. Convert everything to centimetres first, work the missing edge out there, and turn only the final answer back into metres.

3 · Execute4 carry out the plan

1Put the metre edge into centimetres

#8 Analyze the Units 6.RP.A.3

A metre is 100 cm.

3\ \text{m} = 300\ \text{cm}

Two of the three edges now match.

2Multiply the two known edges

#13 Convert to Algebra 6.G.A.2

That is the face the unknown edge stands on.

300 \times 240 = 72000

A face of 72000 cm2.

3Divide the volume by that face

#11 Work Backwards 5.MD.C.5

Volume is base times height, so this recovers the height.

3600000 \div 72000 = 50

The edge is 50 cm.

4Turn it back into metres

#8 Analyze the Units 6.RP.A.3

The label asks for metres, and 100 cm is 1 m.

50\ \text{cm} = 0.5\ \text{m}

The circle holds 0.5.

Answer: 0.5

4 · Reviewdoes it hold up?

Checking forwards: 300 x 240 x 50 = 3600000 cm3, the volume we were given.

Another way: Working in metres throughout needs the volume as 3.6 m3 and the 240 cm edge as 2.4 m -- correct, but with three decimal conversions instead of one.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Relating the prism's volume to its three edges.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Recovering one edge from the volume and a face.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Converting between metres and centimetres.

💡Takeaway. Mixed units cannot be multiplied. Bring them together first, and only change back at the very end.

Variant 2 easy answer: 1.5

The rectangular prism at the right has a volume of $7500000\ \text{cm}^3$ . Find the number that belongs in $\bigcirc$ .

Show solution

1 · Understandwhat's really being asked

A prism has edges 2 m, 250 cm and an unknown number of metres, and its volume is 7500000 cm3. We want the unknown, in metres.

Givens

One edge is 2 m and another is 250 cm.
The volume is 7500000 cm3.
The third edge is given in metres.

Unknowns

The number of metres in the third edge.

Constraints

The volume is in cubic centimetres, so the working has to be in centimetres.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #11 Work Backwards#13 Convert to Algebra

Three lengths in two units and a volume in a third. Convert everything to centimetres first, work the missing edge out there, and turn only the final answer back into metres.

3 · Execute4 carry out the plan

1Put the metre edge into centimetres

#8 Analyze the Units 6.RP.A.3

A metre is 100 cm.

2\ \text{m} = 200\ \text{cm}

Two of the three edges now match.

2Multiply the two known edges

#13 Convert to Algebra 6.G.A.2

That is the face the unknown edge stands on.

200 \times 250 = 50000

A face of 50000 cm2.

3Divide the volume by that face

#11 Work Backwards 5.MD.C.5

Volume is base times height, so this recovers the height.

7500000 \div 50000 = 150

The edge is 150 cm.

4Turn it back into metres

#8 Analyze the Units 6.RP.A.3

The label asks for metres, and 100 cm is 1 m.

150\ \text{cm} = 1.5\ \text{m}

The circle holds 1.5.

Answer: 1.5

4 · Reviewdoes it hold up?

Checking forwards: 200 x 250 x 150 = 7500000 cm3, the volume we were given.

Another way: Working in metres throughout needs the volume as 7.5 m3 and the 250 cm edge as 2.5 m -- correct, but with three decimal conversions instead of one.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Relating the prism's volume to its three edges.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Recovering one edge from the volume and a face.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Converting between metres and centimetres.

💡Takeaway. Mixed units cannot be multiplied. Bring them together first, and only change back at the very end.

Variant 3 easy answer: 0.5

The rectangular prism at the right has a volume of $11200000\ \text{cm}^3$ . Find the number that belongs in $\bigcirc$ .

Show solution

1 · Understandwhat's really being asked

A prism has edges 7 m, 320 cm and an unknown number of metres, and its volume is 11200000 cm3. We want the unknown, in metres.

Givens

One edge is 7 m and another is 320 cm.
The volume is 11200000 cm3.
The third edge is given in metres.

Unknowns

The number of metres in the third edge.

Constraints

The volume is in cubic centimetres, so the working has to be in centimetres.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #11 Work Backwards#13 Convert to Algebra

Three lengths in two units and a volume in a third. Convert everything to centimetres first, work the missing edge out there, and turn only the final answer back into metres.

3 · Execute4 carry out the plan

1Put the metre edge into centimetres

#8 Analyze the Units 6.RP.A.3

A metre is 100 cm.

7\ \text{m} = 700\ \text{cm}

Two of the three edges now match.

2Multiply the two known edges

#13 Convert to Algebra 6.G.A.2

That is the face the unknown edge stands on.

700 \times 320 = 224000

A face of 224000 cm2.

3Divide the volume by that face

#11 Work Backwards 5.MD.C.5

Volume is base times height, so this recovers the height.

11200000 \div 224000 = 50

The edge is 50 cm.

4Turn it back into metres

#8 Analyze the Units 6.RP.A.3

The label asks for metres, and 100 cm is 1 m.

50\ \text{cm} = 0.5\ \text{m}

The circle holds 0.5.

Answer: 0.5

4 · Reviewdoes it hold up?

Checking forwards: 700 x 320 x 50 = 11200000 cm3, the volume we were given.

Another way: Working in metres throughout needs the volume as 11.2 m3 and the 320 cm edge as 3.2 m -- correct, but with three decimal conversions instead of one.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Relating the prism's volume to its three edges.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Recovering one edge from the volume and a face.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Converting between metres and centimetres.

💡Takeaway. Mixed units cannot be multiplied. Bring them together first, and only change back at the very end.

Variant 4 easy answer: 3.5

The rectangular prism at the right has a volume of $31500000\ \text{cm}^3$ . Find the number that belongs in $\bigcirc$ .

Show solution

1 · Understandwhat's really being asked

A prism has edges 2 m, 450 cm and an unknown number of metres, and its volume is 31500000 cm3. We want the unknown, in metres.

Givens

One edge is 2 m and another is 450 cm.
The volume is 31500000 cm3.
The third edge is given in metres.

Unknowns

The number of metres in the third edge.

Constraints

The volume is in cubic centimetres, so the working has to be in centimetres.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #11 Work Backwards#13 Convert to Algebra

Three lengths in two units and a volume in a third. Convert everything to centimetres first, work the missing edge out there, and turn only the final answer back into metres.

3 · Execute4 carry out the plan

1Put the metre edge into centimetres

#8 Analyze the Units 6.RP.A.3

A metre is 100 cm.

2\ \text{m} = 200\ \text{cm}

Two of the three edges now match.

2Multiply the two known edges

#13 Convert to Algebra 6.G.A.2

That is the face the unknown edge stands on.

200 \times 450 = 90000

A face of 90000 cm2.

3Divide the volume by that face

#11 Work Backwards 5.MD.C.5

Volume is base times height, so this recovers the height.

31500000 \div 90000 = 350

The edge is 350 cm.

4Turn it back into metres

#8 Analyze the Units 6.RP.A.3

The label asks for metres, and 100 cm is 1 m.

350\ \text{cm} = 3.5\ \text{m}

The circle holds 3.5.

Answer: 3.5

4 · Reviewdoes it hold up?

Checking forwards: 200 x 450 x 350 = 31500000 cm3, the volume we were given.

Another way: Working in metres throughout needs the volume as 31.5 m3 and the 450 cm edge as 4.5 m -- correct, but with three decimal conversions instead of one.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Relating the prism's volume to its three edges.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Recovering one edge from the volume and a face.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Converting between metres and centimetres.

💡Takeaway. Mixed units cannot be multiplied. Bring them together first, and only change back at the very end.

Variant 5 medium answer: 5.5

The rectangular prism at the right has a volume of $49500000\ \text{cm}^3$ . Find the number that belongs in $\bigcirc$ .

Show solution

1 · Understandwhat's really being asked

A prism has edges 2 m, 450 cm and an unknown number of metres, and its volume is 49500000 cm3. We want the unknown, in metres.

Givens

One edge is 2 m and another is 450 cm.
The volume is 49500000 cm3.
The third edge is given in metres.

Unknowns

The number of metres in the third edge.

Constraints

The volume is in cubic centimetres, so the working has to be in centimetres.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #11 Work Backwards#13 Convert to Algebra

Three lengths in two units and a volume in a third. Convert everything to centimetres first, work the missing edge out there, and turn only the final answer back into metres.

3 · Execute4 carry out the plan

1Put the metre edge into centimetres

#8 Analyze the Units 6.RP.A.3

A metre is 100 cm.

2\ \text{m} = 200\ \text{cm}

Two of the three edges now match.

2Multiply the two known edges

#13 Convert to Algebra 6.G.A.2

That is the face the unknown edge stands on.

200 \times 450 = 90000

A face of 90000 cm2.

3Divide the volume by that face

#11 Work Backwards 5.MD.C.5

Volume is base times height, so this recovers the height.

49500000 \div 90000 = 550

The edge is 550 cm.

4Turn it back into metres

#8 Analyze the Units 6.RP.A.3

The label asks for metres, and 100 cm is 1 m.

550\ \text{cm} = 5.5\ \text{m}

The circle holds 5.5.

Answer: 5.5

4 · Reviewdoes it hold up?

Checking forwards: 200 x 450 x 550 = 49500000 cm3, the volume we were given.

Another way: Working in metres throughout needs the volume as 49.5 m3 and the 450 cm edge as 4.5 m -- correct, but with three decimal conversions instead of one.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Relating the prism's volume to its three edges.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Recovering one edge from the volume and a face.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Converting between metres and centimetres.

💡Takeaway. Mixed units cannot be multiplied. Bring them together first, and only change back at the very end.

Variant 6 medium answer: 6.5

The rectangular prism at the right has a volume of $58500000\ \text{cm}^3$ . Find the number that belongs in $\bigcirc$ .

Show solution

1 · Understandwhat's really being asked

A prism has edges 6 m, 150 cm and an unknown number of metres, and its volume is 58500000 cm3. We want the unknown, in metres.

Givens

One edge is 6 m and another is 150 cm.
The volume is 58500000 cm3.
The third edge is given in metres.

Unknowns

The number of metres in the third edge.

Constraints

The volume is in cubic centimetres, so the working has to be in centimetres.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #11 Work Backwards#13 Convert to Algebra

Three lengths in two units and a volume in a third. Convert everything to centimetres first, work the missing edge out there, and turn only the final answer back into metres.

3 · Execute4 carry out the plan

1Put the metre edge into centimetres

#8 Analyze the Units 6.RP.A.3

A metre is 100 cm.

6\ \text{m} = 600\ \text{cm}

Two of the three edges now match.

2Multiply the two known edges

#13 Convert to Algebra 6.G.A.2

That is the face the unknown edge stands on.

600 \times 150 = 90000

A face of 90000 cm2.

3Divide the volume by that face

#11 Work Backwards 5.MD.C.5

Volume is base times height, so this recovers the height.

58500000 \div 90000 = 650

The edge is 650 cm.

4Turn it back into metres

#8 Analyze the Units 6.RP.A.3

The label asks for metres, and 100 cm is 1 m.

650\ \text{cm} = 6.5\ \text{m}

The circle holds 6.5.

Answer: 6.5

4 · Reviewdoes it hold up?

Checking forwards: 600 x 150 x 650 = 58500000 cm3, the volume we were given.

Another way: Working in metres throughout needs the volume as 58.5 m3 and the 150 cm edge as 1.5 m -- correct, but with three decimal conversions instead of one.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Relating the prism's volume to its three edges.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Recovering one edge from the volume and a face.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Converting between metres and centimetres.

💡Takeaway. Mixed units cannot be multiplied. Bring them together first, and only change back at the very end.

Variant 7 medium answer: 7.5

The rectangular prism at the right has a volume of $67500000\ \text{cm}^3$ . Find the number that belongs in $\bigcirc$ .

Show solution

1 · Understandwhat's really being asked

A prism has edges 2 m, 450 cm and an unknown number of metres, and its volume is 67500000 cm3. We want the unknown, in metres.

Givens

One edge is 2 m and another is 450 cm.
The volume is 67500000 cm3.
The third edge is given in metres.

Unknowns

The number of metres in the third edge.

Constraints

The volume is in cubic centimetres, so the working has to be in centimetres.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #11 Work Backwards#13 Convert to Algebra

Three lengths in two units and a volume in a third. Convert everything to centimetres first, work the missing edge out there, and turn only the final answer back into metres.

3 · Execute4 carry out the plan

1Put the metre edge into centimetres

#8 Analyze the Units 6.RP.A.3

A metre is 100 cm.

2\ \text{m} = 200\ \text{cm}

Two of the three edges now match.

2Multiply the two known edges

#13 Convert to Algebra 6.G.A.2

That is the face the unknown edge stands on.

200 \times 450 = 90000

A face of 90000 cm2.

3Divide the volume by that face

#11 Work Backwards 5.MD.C.5

Volume is base times height, so this recovers the height.

67500000 \div 90000 = 750

The edge is 750 cm.

4Turn it back into metres

#8 Analyze the Units 6.RP.A.3

The label asks for metres, and 100 cm is 1 m.

750\ \text{cm} = 7.5\ \text{m}

The circle holds 7.5.

Answer: 7.5

4 · Reviewdoes it hold up?

Checking forwards: 200 x 450 x 750 = 67500000 cm3, the volume we were given.

Another way: Working in metres throughout needs the volume as 67.5 m3 and the 450 cm edge as 4.5 m -- correct, but with three decimal conversions instead of one.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Relating the prism's volume to its three edges.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Recovering one edge from the volume and a face.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Converting between metres and centimetres.

💡Takeaway. Mixed units cannot be multiplied. Bring them together first, and only change back at the very end.

Variant 8 medium answer: 8.5

The rectangular prism at the right has a volume of $68000000\ \text{cm}^3$ . Find the number that belongs in $\bigcirc$ .

Show solution

1 · Understandwhat's really being asked

A prism has edges 4 m, 200 cm and an unknown number of metres, and its volume is 68000000 cm3. We want the unknown, in metres.

Givens

One edge is 4 m and another is 200 cm.
The volume is 68000000 cm3.
The third edge is given in metres.

Unknowns

The number of metres in the third edge.

Constraints

The volume is in cubic centimetres, so the working has to be in centimetres.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #11 Work Backwards#13 Convert to Algebra

Three lengths in two units and a volume in a third. Convert everything to centimetres first, work the missing edge out there, and turn only the final answer back into metres.

3 · Execute4 carry out the plan

1Put the metre edge into centimetres

#8 Analyze the Units 6.RP.A.3

A metre is 100 cm.

4\ \text{m} = 400\ \text{cm}

Two of the three edges now match.

2Multiply the two known edges

#13 Convert to Algebra 6.G.A.2

That is the face the unknown edge stands on.

400 \times 200 = 80000

A face of 80000 cm2.

3Divide the volume by that face

#11 Work Backwards 5.MD.C.5

Volume is base times height, so this recovers the height.

68000000 \div 80000 = 850

The edge is 850 cm.

4Turn it back into metres

#8 Analyze the Units 6.RP.A.3

The label asks for metres, and 100 cm is 1 m.

850\ \text{cm} = 8.5\ \text{m}

The circle holds 8.5.

Answer: 8.5

4 · Reviewdoes it hold up?

Checking forwards: 400 x 200 x 850 = 68000000 cm3, the volume we were given.

Another way: Working in metres throughout needs the volume as 68 m3 and the 200 cm edge as 2 m -- correct, but with three decimal conversions instead of one.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Relating the prism's volume to its three edges.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Recovering one edge from the volume and a face.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Converting between metres and centimetres.

💡Takeaway. Mixed units cannot be multiplied. Bring them together first, and only change back at the very end.

Variant 9 hard answer: 6.5

The rectangular prism at the right has a volume of $72800000\ \text{cm}^3$ . Find the number that belongs in $\bigcirc$ .

Show solution

1 · Understandwhat's really being asked

A prism has edges 7 m, 160 cm and an unknown number of metres, and its volume is 72800000 cm3. We want the unknown, in metres.

Givens

One edge is 7 m and another is 160 cm.
The volume is 72800000 cm3.
The third edge is given in metres.

Unknowns

The number of metres in the third edge.

Constraints

The volume is in cubic centimetres, so the working has to be in centimetres.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #11 Work Backwards#13 Convert to Algebra

Three lengths in two units and a volume in a third. Convert everything to centimetres first, work the missing edge out there, and turn only the final answer back into metres.

3 · Execute4 carry out the plan

1Put the metre edge into centimetres

#8 Analyze the Units 6.RP.A.3

A metre is 100 cm.

7\ \text{m} = 700\ \text{cm}

Two of the three edges now match.

2Multiply the two known edges

#13 Convert to Algebra 6.G.A.2

That is the face the unknown edge stands on.

700 \times 160 = 112000

A face of 112000 cm2.

3Divide the volume by that face

#11 Work Backwards 5.MD.C.5

Volume is base times height, so this recovers the height.

72800000 \div 112000 = 650

The edge is 650 cm.

4Turn it back into metres

#8 Analyze the Units 6.RP.A.3

The label asks for metres, and 100 cm is 1 m.

650\ \text{cm} = 6.5\ \text{m}

The circle holds 6.5.

Answer: 6.5

4 · Reviewdoes it hold up?

Checking forwards: 700 x 160 x 650 = 72800000 cm3, the volume we were given.

Another way: Working in metres throughout needs the volume as 72.8 m3 and the 160 cm edge as 1.6 m -- correct, but with three decimal conversions instead of one.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Relating the prism's volume to its three edges.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Recovering one edge from the volume and a face.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Converting between metres and centimetres.

💡Takeaway. Mixed units cannot be multiplied. Bring them together first, and only change back at the very end.

Variant 10 hard answer: 7.5

The rectangular prism at the right has a volume of $112500000\ \text{cm}^3$ . Find the number that belongs in $\bigcirc$ .

Show solution

1 · Understandwhat's really being asked

A prism has edges 3 m, 500 cm and an unknown number of metres, and its volume is 112500000 cm3. We want the unknown, in metres.

Givens

One edge is 3 m and another is 500 cm.
The volume is 112500000 cm3.
The third edge is given in metres.

Unknowns

The number of metres in the third edge.

Constraints

The volume is in cubic centimetres, so the working has to be in centimetres.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #11 Work Backwards#13 Convert to Algebra

Three lengths in two units and a volume in a third. Convert everything to centimetres first, work the missing edge out there, and turn only the final answer back into metres.

3 · Execute4 carry out the plan

1Put the metre edge into centimetres

#8 Analyze the Units 6.RP.A.3

A metre is 100 cm.

3\ \text{m} = 300\ \text{cm}

Two of the three edges now match.

2Multiply the two known edges

#13 Convert to Algebra 6.G.A.2

That is the face the unknown edge stands on.

300 \times 500 = 150000

A face of 150000 cm2.

3Divide the volume by that face

#11 Work Backwards 5.MD.C.5

Volume is base times height, so this recovers the height.

112500000 \div 150000 = 750

The edge is 750 cm.

4Turn it back into metres

#8 Analyze the Units 6.RP.A.3

The label asks for metres, and 100 cm is 1 m.

750\ \text{cm} = 7.5\ \text{m}

The circle holds 7.5.

Answer: 7.5

4 · Reviewdoes it hold up?

Checking forwards: 300 x 500 x 750 = 112500000 cm3, the volume we were given.

Another way: Working in metres throughout needs the volume as 112.5 m3 and the 500 cm edge as 5 m -- correct, but with three decimal conversions instead of one.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Relating the prism's volume to its three edges.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Recovering one edge from the volume and a face.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Converting between metres and centimetres.

💡Takeaway. Mixed units cannot be multiplied. Bring them together first, and only change back at the very end.

Variant 11 hard answer: 12.5

The rectangular prism at the right has a volume of $187500000\ \text{cm}^3$ . Find the number that belongs in $\bigcirc$ .

Show solution

1 · Understandwhat's really being asked

A prism has edges 3 m, 500 cm and an unknown number of metres, and its volume is 187500000 cm3. We want the unknown, in metres.

Givens

One edge is 3 m and another is 500 cm.
The volume is 187500000 cm3.
The third edge is given in metres.

Unknowns

The number of metres in the third edge.

Constraints

The volume is in cubic centimetres, so the working has to be in centimetres.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #11 Work Backwards#13 Convert to Algebra

Three lengths in two units and a volume in a third. Convert everything to centimetres first, work the missing edge out there, and turn only the final answer back into metres.

3 · Execute4 carry out the plan

1Put the metre edge into centimetres

#8 Analyze the Units 6.RP.A.3

A metre is 100 cm.

3\ \text{m} = 300\ \text{cm}

Two of the three edges now match.

2Multiply the two known edges

#13 Convert to Algebra 6.G.A.2

That is the face the unknown edge stands on.

300 \times 500 = 150000

A face of 150000 cm2.

3Divide the volume by that face

#11 Work Backwards 5.MD.C.5

Volume is base times height, so this recovers the height.

187500000 \div 150000 = 1250

The edge is 1250 cm.

4Turn it back into metres

#8 Analyze the Units 6.RP.A.3

The label asks for metres, and 100 cm is 1 m.

1250\ \text{cm} = 12.5\ \text{m}

The circle holds 12.5.

Answer: 12.5

4 · Reviewdoes it hold up?

Checking forwards: 300 x 500 x 1250 = 187500000 cm3, the volume we were given.

Another way: Working in metres throughout needs the volume as 187.5 m3 and the 500 cm edge as 5 m -- correct, but with three decimal conversions instead of one.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Relating the prism's volume to its three edges.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Recovering one edge from the volume and a face.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Converting between metres and centimetres.

💡Takeaway. Mixed units cannot be multiplied. Bring them together first, and only change back at the very end.

Variant 12 hard answer: 7.5

The rectangular prism at the right has a volume of $236250000\ \text{cm}^3$ . Find the number that belongs in $\bigcirc$ .

Show solution

1 · Understandwhat's really being asked

A prism has edges 7 m, 450 cm and an unknown number of metres, and its volume is 236250000 cm3. We want the unknown, in metres.

Givens

One edge is 7 m and another is 450 cm.
The volume is 236250000 cm3.
The third edge is given in metres.

Unknowns

The number of metres in the third edge.

Constraints

The volume is in cubic centimetres, so the working has to be in centimetres.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #11 Work Backwards#13 Convert to Algebra

Three lengths in two units and a volume in a third. Convert everything to centimetres first, work the missing edge out there, and turn only the final answer back into metres.

3 · Execute4 carry out the plan

1Put the metre edge into centimetres

#8 Analyze the Units 6.RP.A.3

A metre is 100 cm.

7\ \text{m} = 700\ \text{cm}

Two of the three edges now match.

2Multiply the two known edges

#13 Convert to Algebra 6.G.A.2

That is the face the unknown edge stands on.

700 \times 450 = 315000

A face of 315000 cm2.

3Divide the volume by that face

#11 Work Backwards 5.MD.C.5

Volume is base times height, so this recovers the height.

236250000 \div 315000 = 750

The edge is 750 cm.

4Turn it back into metres

#8 Analyze the Units 6.RP.A.3

The label asks for metres, and 100 cm is 1 m.

750\ \text{cm} = 7.5\ \text{m}

The circle holds 7.5.

Answer: 7.5

4 · Reviewdoes it hold up?

Checking forwards: 700 x 450 x 750 = 236250000 cm3, the volume we were given.

Another way: Working in metres throughout needs the volume as 236.25 m3 and the 450 cm edge as 4.5 m -- correct, but with three decimal conversions instead of one.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Relating the prism's volume to its three edges.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Recovering one edge from the volume and a face.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Converting between metres and centimetres.

💡Takeaway. Mixed units cannot be multiplied. Bring them together first, and only change back at the very end.